Different things have various life phases. A novel or short story, for example, generally follow five phases as shown in the image below.
An MMORPG too, has different phases. (NOTE: This is not official. I just made this according to what I have observed about the industry) It starts out with the building of hype. The game isn't out yet, but the company is doing its best to get publicity. In this phase, we see trailer videos, character information, etc. This is where players build expectations about the game.
The next phase can be its CBT, OBT, or launch. Once the game is opened, thousands of players will rush to log-in. The first few days of the game is probably the peak of any game. This is generally true, except on special occasions when the game slowly gains popularity once it is already open (perfect example would be WOW).
After a few weeks, we will enter stage 3. Some of the players will leave since the game did not live up to their expectations. Others become bored of it. Still others, will temporarily leave and wait for further updates. The reason why the starting days of an MMORPG are usually the peak of the game is that, although new players can still try the game at a latter point, more players have stopped playing altogether. The efflux will generally be higher than the influx in this stage.
Stage 4 is when the game matures. The game will continuously lose a few players, but is cushioned by the influx of few new players. At this point, players who are really loving the game and are loyal to the game are the ones who are primarily making the community alive.
At this stage, three things can happen. The game can stabilize its population by keeping its loyal player base and making events that would give players a reason to stay. The second thing is that it will continue to lose players until it closes. The third one, is demonstrated in the image where a game will experience another increase in player base. This is stage 5.
Stage 5 is when players start to come in again due to one of the following reasons: new content (level cap increase, new zone, new character/class, etc.), major overhaul that removes features that players don't want and adds features that players want, and/or any major changes for the better (sometimes for the worse but people will try then leave soon after they discover it).
The last stage is when the game faces the inevitable. It is the time when no more content update is being done, or when no content update could save the game. Stages 4 and 5 can happen several times in a game's lifetime especially if it gets frequent updates. However, the game will definitely die. The number of months or years may vary per game, but it has to close down eventually.
This is what I think a typical MMORPG goes through. If you see any deviations, feel free to comment. If you know of MMORPGs that don't follow this layout, feel free to write it down. I'm sure there are MMORPGs that don't follow this, but I think this is generally the case for many MMOs.
Lastly, I would like to state the obvious. Nothing lasts forever. If WOW, for example, is starting to experience a decline, it's only natural. Fanboys and haters will debate on it, but it's just a reality all games follow. Everything dies, even WOW (when it will happen, though, nobody knows). XD